5 Many business profits can be modeled by quadratic functionsMany business profits can be modeled by quadratic functions. To ensure that the profit is above a certain level, financial planners may need to graph and solve quadratic inequalities.A quadratic inequality in two variables can be written in one of the following forms, where a, b, and c are real numbers and a ≠ 0. Its solution set is a set of ordered pairs (x, y).

16 Quadratic inequalities in one variable, such as ax2 + bx + c > 0 (a ≠ 0), have solutions in one variable that are graphed on a number line.For and statements, both of the conditions must be true. For or statements, at least one of the conditions must be true.Reading Math

17 Example 2A: Solving Quadratic Inequalities by Using Tables and GraphsSolve the inequality by using tables or graphs.x2 + 8x + 20 ≥ 5Use a graphing calculator to graph each side of the inequality. Set Y1 equal to x2 + 8x + 20 and Y2 equal to 5. Identify the values of x for which Y1 ≥ Y2.

18 The number line shows the solution set.Example 2A ContinuedThe parabola is at or above the line when x is less than or equal to –5 or greater than or equal to –3. So, the solution set is x ≤ –5 or x ≥ –3 or (–∞, –5] U [–3, ∞). The table supports your answer.The number line shows the solution set.–6 –4 –

19 Example 2B: Solving Quadratics Inequalities by Using Tables and GraphsSolve the inequality by using tables and graph.x2 + 8x + 20 < 5Use a graphing calculator to graph each side of the inequality. Set Y1 equal to x2 + 8x + 20 and Y2 equal to 5. Identify the values of which Y1 < Y2.

20 The number line shows the solution set.Example 2B ContinuedThe parabola is below the line when x is greater than –5 and less than –3. So, the solution set is –5 < x < –3 or (–5, –3). The table supports your answer.The number line shows the solution set.–6 –4 –

21 Check It Out! Example 2aSolve the inequality by using tables and graph.x2 – x + 5 < 7Use a graphing calculator to graph each side of the inequality. Set Y1 equal to x2 – x + 5 and Y2 equal to 7. Identify the values of which Y1 < Y2.

22 Check It Out! Example 2a ContinuedThe parabola is below the line when x is greater than –1 and less than 2. So, the solution set is –1 < x < 2 or (–1, 2). The table supports your answer.The number line shows the solution set.–6 –4 –

23 Check It Out! Example 2bSolve the inequality by using tables and graph.2x2 – 5x + 1 ≥ 1Use a graphing calculator to graph each side of the inequality. Set Y1 equal to 2x2 – 5x + 1 and Y2 equal to 1. Identify the values of which Y1 ≥ Y2.

24 Check It Out! Example 2b ContinuedThe parabola is at or above the line when x is less than or equal to 0 or greater than or greater than or equal to 2.5. So, the solution set is (–∞, 0] U [2.5, ∞)The number line shows the solution set.–6 –4 –

25 The number lines showing the solution sets in Example 2 are divided into three distinct regions by the points –5 and –3. These points are called critical values. By finding the critical values, you can solve quadratic inequalities algebraically.

33 Check It Out! Example 3a ContinuedShade the solution regions on the number line. Use solid circles for the critical values because the inequality contains them. The solution is x ≤ 2 or x ≥ 4.–3 –2 –

37 Check It Out! Example 3Shade the solution regions on the number line. Use open circles for the critical values because the inequality does not contain or equal to. The solution is x < –1 or x > 2.5.–3 –2 –

39 Example 4: Problem-Solving ApplicationThe monthly profit P of a small business that sells bicycle helmets can be modeled by the function P(x) = –8x x – 4200, where x is the average selling price of a helmet. What range of selling prices will generate a monthly profit of at least $6000?

40 Understand the ProblemExample 4 Continued1Understand the ProblemThe answer will be the average price of a helmet required for a profit that is greater than or equal to $6000.List the important information:The profit must be at least $6000.The function for the business’s profit is P(x) = –8x x – 4200.

41 Example 4 Continued2Make a PlanWrite an inequality showing profit greater than or equal to $6000. Then solve the inequality by using algebra.

46 Example 4 ContinuedSolve3For a profit of $6000, the average price of a helmet needs to be between $26.04 and $48.96, inclusive.

47 Example 4 ContinuedLook Back4Enter y = –8x x – 4200 into a graphing calculator, and create a table of values. The table shows that integer values of x between and inclusive result in y-values greater than or equal to 6000.

48 Check It Out! Example 4A business offers educational tours to Patagonia, a region of South America that includes parts of Chile and Argentina . The profit P for x number of persons is P(x) = –25x x – The trip will be rescheduled if the profit is less $7500. How many people must have signed up if the trip is rescheduled?

49 Understand the ProblemCheck It Out! Example 4 Continued1Understand the ProblemThe answer will be the number of people signed up for the trip if the profit is less than $7500.List the important information:The profit will be less than $7500.The function for the profit is P(x) = –25x x – 5000.

50 Check It Out! Example 4 Continued2Make a PlanWrite an inequality showing profit less than $7500. Then solve the inequality by using algebra.

55 Check It Out! Example 4 ContinuedSolve3The trip will be rescheduled if the number of people signed up is fewer than 14 people or more than 36 people.

56 Check It Out! Example 4 ContinuedLook Back4Enter y = –25x x – 5000 into a graphing calculator, and create a table of values. The table shows that integer values of x less than and greater than result in y-values less than 7500.

58 Lesson Quiz: Part II4. A boat operator wants to offer tours of San Francisco Bay. His profit P for a trip can be modeled by P(x) = –2x x – 788, where x is the cost per ticket. What range of ticket prices will generate a profit of at least $500?between $14 and $46, inclusive